The CM conjecture for arithmetic shtukas with one paw

Let $C$ be the ambient complete algebraically closed field, and let $G=(V,\mathcal{L})$ be a shtuka with one paw: $V$ is a finite-dimensional $\mathbb{Q}_p$-vector space and $\mathcal{L}\subset V\otimes\mathbb{B}_{\mathrm{dR}}$ is a $\mathbb{B}_{\mathrm{dR}}^+$-lattice. The Hodge–Tate filtration is the filtration on $V\otimes C$ induced by $\mathcal{L}$ via $\theta:\mathbb{B}_{\mathrm{dR}}^+\to C$. The shtuka is arithmetic if it is isomorphic to one arising from a complete discretely valued subfield $L\subset C$ and a de Rham representation of $\operatorname{Gal}(\overline{L}/L)$, and it has CM if it admits endomorphisms by a semisimple commutative $\mathbb{Q}_p$-algebra of rank $\dim V$. The CM conjecture. If $G$ is an arithmetic shtuka with one paw whose Hodge–Tate filtration is defined over a complete discretely valued subfield of $C$, then $G$ has CM. This is an optimistic generalization of the transcendence result for one-dimensional $p$-divisible groups to shtukas with one paw; the paper presents it as a conjectural extension, and no resolution is supplied here.